Random-Cluster Representation for the Blume–Capel Model

We present in this paper a way to perform the mapping of the spin-1 Blume–Capel model into a random-cluster model, and analyze thermodynamic properties of the former model in terms of geometric properties of clusters generated in the random-cluster representation. It is shown that there are two different relevant types of cluster, and that one of them is the exact analogue of the type of cluster generated in the Ising model. We use this result to derive expressions for thermodynamical properties on the second-order transition line which are equivalent to the ones found in the Ising model. The other type of cluster is responsible for the first-order transitions, and we may see the tricritical point as a point where both types of cluster compete on the same footing.